In Section  we prove compactness and lower-semicontinuity properties for a sequence of \(m\) -minimizers. We begin by establishing in Proposition REF  the compactness for a sequence of \(m\) -minimizers \(\lbrace (A_m,u_m)\rbrace \)  with free-crystal regions \(A_m\)  not containing isolated points of such free-crystal regions to a limiting set of finite perimeter \(A\subset \Omega \)  by means of both the Blaschke-type selection principle [1]} and the density estimates established in Section . Then, in Proposition REF , we further extend the (already generalized) Golab-type Theorem [2]} to a priori not-connected \(\mathcal {H}^1\) -measurable (not necessarily \(\mathcal {H}^1\) -rectifiable) sets satisfying uniform density estimates (see [3]} for the isotropic case). The compactness of the displacements in \(\lbrace (A_m,u_m)\rbrace \)  is then proved in Propositions REF  by carefully constructing the limiting displacement \(u\)  in view of the property that for every connected component \(E_i\)  of \(A\)  the set in which displacements \(u_m\)  diverge is either the whole component \(E_i\)  or \(\emptyset \) , which follows from [1]}.
Finally, in Proposition REF  we establish the lower-semicontinuity property
\(\liminf \limits _{h\rightarrow \infty } \mathcal {F}(A_{m_h},u_{m_h}) \ge \widetilde{\mathcal {F}}(A,u),\) 
